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Mathematical examples. Solve nonlinear algebraic equations. Find minimum of a nonlinear function. Determine eigenvalues

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18-660: Numerical Methods for Engineering Design and Optimization Xin Li Department of ECE Carnegie Mellon University Pittsburgh, PA 15213

Slide 1

Overview 

Introduction Numerical computation  Examples and applications 

Slide 2

Numerical Computation 

Numerical computation: approximation techniques and algorithms to numerically solve mathematical problems 



Not all mathematical problems have closed-form solutions

Mathematical examples Solve nonlinear algebraic equations  Find minimum of a nonlinear function  Determine eigenvalues of a matrix 



This course will encompass various aspects of numerical methods and their applications 

Linear & nonlinear solver, nonlinear optimization, randomized algorithm, etc. Slide 3

Numerical Computation 

Numerical computation is an important tool to solve many practical engineering problems PageRank finds the eigenvector of a 1010-by-1010 matrix – the largest matrix computation in the world

Airlines use optimization algorithms to decide ticket prices, airplane assignments and fuel needs

Slide 4

Numerical Computation 

Several additional real-world examples: Car companies run computer simulations of car crashes to identify safety issues

Wall street runs statistical simulation tools to predict stock prices

Slide 5

Brief History of Numerical Computation 

Before 1950s



Algorithms Linear interpolation  Newton’s method  Gaussian elimination 



Tools Hand calculation  Formula handbook  Data table 

Carl Friedrich Gauss (1777-1855) German Mathematician

Slide 6

Brief History of Numerical Computation After 1950s 

The invention of modern computers motivates the development of a great number of new numerical algorithms 1E+11

1E+10 1E+09

1E+09 1E+08

1E+08

1E+07

1E+07

1E+06 1E+06

CPU Frequency (Hz)

1E+10

1E+05

IPS

2008

2007

2006

2006

2005

2005

2002

2000

1999

1996

1994

1990

1986

1984

1979

1E+05

1974

1E+04

1971

Instruction Per Second



Freq

Slide 7

Numerical Computation Problems 

Ordinary and partial differential equations



Regression



Classification

Slide 8

Ordinary Differential Equation (ODE) 

Transient analysis for electrical circuit u (t ) dv(t ) = u (t ) − v(t ) dt dv(t ) v(t ) − v(t − ∆t ) ≈ dt ∆t

v(t )

1

1

1Ω

0

1F

0

t5

t6

v(t)

v(t1 ) − v(t0 ) = u (t1 ) − v(t1 ) ∆t v(t 2 ) − v(t1 ) = u (t 2 ) − v(t 2 ) ∆t 

t0

t1

t2

t3

t4

t7

t

Algebraic equations Slide 9

Partial Differential Equation (PDE) 

Thermal analysis: heat conduction is governed by partial differential equation (PDE): ρ ⋅Cp ⋅

Material density (kg/m3)

∂T ( x, y, z , t ) ∂t

Specific heat capacity (Joules/K-kg)

= ∇ ⋅ [κ ⋅ ∇T ( x, y, z , t )] + p( x, y, z , t ) Thermal conductivity (Watts/K-m)

Power density of heat sources (Watts/m3)

Slide 10

Linear Regression 

Linear regression (response surface modeling) f ( x ) = ax + b

f(x)

f ( x1 ) = ax1 + b

f ( x2 ) = ax2 + b f ( x3 ) = ax3 + b 

x

Solve over-determined linear equation to find a and b (more equations than unknowns)

Slide 11

Linear Regression 

Linear regression with regularization Aα = B

min α

Aα − B 2 + λ ⋅ α 2

1

Least-squares L1-norm error regularization

Original image

Recovered image Slide 12

Classification Support vector machine (SVM) f (X ) = W X + C T

≥ 0  < 0

SVM Coefficients Features

(Class A) (Class B )

Feature x2



min W 2 L2-norm (maximize margin) S.T. W T X + C ≥ 0 ( X ∈ Class A) W T X + C < 0 ( X ∈ Class B ) 2

Margin Feature x1

Solve convex quadratic programming to find W and C Slide 13

Classification Brain computer interface based on classification MEG

2

2

1.5

1.5

1

Amplitude (Normalized)

EEG

Amplitude (Normalized)



0.5 0 -0.5 -1 -1.5 -2 -2.5 -0.5

1 0.5 0 -0.5 -1 -1.5 -2

0

0.5 1 Time (Sec.)

1.5

2

-2.5 -0.5

0

0.5 1 Time (Sec.)

1.5

2

Methods: filtering; signal subspace methods … Tool Boxes: MNE-Suite; FieldTrip; MaxFilter …

Neural signal recording

Neural signal processing

Movement decoding

Feedback Slide 14

Summary  

Numerical computation Examples and applications

Slide 15

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